Boundary values of diffeomorphisms of simple polytopes, and controllability

Let M be a convex polytope with non-empty interior in a finite dimensional vector space, such that each vertex of M is contained in exactly n edges of M. The Lie group of all smooth diffeomorphisms of M contains the Lie subgroup of all diffeomorphisms which restrict to the identity on the boundary of the polytope. We give the quotient of the diffeomorphism group mod subgroup a Lie group structure. This construction turns the canonical quotient homomorphism into a smooth submersion, and we then obtain related results. We also study controllability on M and show that the identity component of the diffeomorphism group is generated by the exponential image.